Let $\Delta ABC$ be a triangle. The internal bisectors of angles $\angle CAB$ and $\angle ABC$ intersect segments $BC$, respectively $AC$ in $D$, respectively $E$. Prove that \[DE\leq (3-2\sqrt{2})(AB+BC+CA).\]

Let $f(x)=ax^2+bx+c$ where $a,b,c$ are real numbers. Suppose $f(-1),f(0),f(1) \in [-1,1]$. Prove that $|f(x)|\le \frac{3}{2}$ for all $x \in [-1,1]$.

Two subsets $A$ and $B$ of the $(x,y)$-plane are said to be [i]equivalent[/i] if there exists a function $f: A\to B$ which is both one-to-one and onto. (i) Show that any two line segments in the plane are equivalent. (ii) Show that any two circles in the plane are equivalent.

Let $x_1, x_2, \ldots , x_n \ (n \geq 1)$ be real numbers such that $0 \leq x_j \leq \pi, \ j = 1, 2,\ldots, n.$ Prove that if $\sum_{j=1}^n (\cos x_j +1) $ is an odd integer, then $\sum_{j=1}^n \sin x_j \geq 1.$

(a) Find out, whether there exists a differentiable function $f: R \longrightarrow R$, not equaling $0$ for all $x \in R$, satisfying the conditions $2f(f(x)) = f(x) \geq 0$ for all $x \in R$. (b) Find out, whether there exists a differentiable function $f: R \longrightarrow R$, not equaling $0$ for all $x \in R$, satisfying the conditions $-1 \leq 2f(f(x)) = f(x) \leq 1$ for all $x \in R$.

Let $a$ be real number with $\sqrt{2}<a<2$, and let $ABCD$ be a convex cyclic quadrilateral whose circumcentre $O$ lies in its interior. The quadrilateral's circumcircle $\omega$ has radius $1$, and the longest and shortest sides of the quadrilateral have length $a$ and $\sqrt{4-a^2}$, respectively. Lines $L_A,L_B,L_C,L_D$ are tangent to $\omega$ at $A,B,C,D$, respectively. Let lines $L_A$ and $L_B$, $L_B$ and $L_C$,$L_C$ and $L_D$,$L_D$ and $L_A$ intersect at $A',B',C',D'$ respectively. Determine the minimum value of $\frac{S_{A'B'C'D'}}{S_{ABCD}}$.

Let $ S $ be a set of $ n $ concentric circles in the plane. Prove that if a function $ f: S\to S $ satisfies the property \[ d( f(A),f(B)) \geq d(A,B) \] for all $ A,B \in S $, then $ d(f(A),f(B)) = d(A,B) $, where $ d $ is the euclidean distance function.

Let $f(x)=\sqrt{\sin^4 x + 4\cos^2 x}-\sqrt{\cos^4x + 4\sin^2x}$. An equivalent form of $f(x)$ is $\textbf{(A) }1-\sqrt2\sin x\qquad\textbf{(B) }-1+\sqrt2\cos x\qquad\textbf{(C) }\cos\dfrac x2-\sin\dfrac x2$ $\textbf{(D) }\cos x-\sin x\qquad\textbf{(E) }\cos2x$

Given three positive real numbers $ x$, $ y$, $ z$ such that $ x \plus{} y \plus{} z \equal{} 1$, prove that $ \frac {xy}{\sqrt {xy \plus{} yz}} \plus{} \frac {yz}{\sqrt {yz \plus{} zx}} \plus{} \frac {zx}{\sqrt {zx \plus{} xy}} \le \frac {\sqrt {2}}{2}$.

Call a real-valued function $ f$ [i]very convex[/i] if \[ \frac {f(x) \plus{} f(y)}{2} \ge f\left(\frac {x \plus{} y}{2}\right) \plus{} |x \minus{} y| \] holds for all real numbers $ x$ and $ y$. Prove that no very convex function exists.

Consider an integrable function $f:\mathbb{R} \rightarrow \mathbb{R}$ such that $af(a)+bf(b)=0$ when $ab=1$. Find the value of the following integration: $$ \int_{0}^{\infty} f(x) \,dx $$

For every integer $n>2$, prove the equality $$\left\lfloor\frac{n(n+1)}{4n-2}\right\rfloor=\left\lfloor\frac{n+1}4\right\rfloor.$$

Find all functions $f$ and $g$ defined from $\mathbb{R}_{>0}$ to $\mathbb{R}_{>0}$ such that for all $x, y > 0$ the two equations hold $$ (f(x) + y - 1)(g(y) + x - 1) = {(x + y)}^2 $$ $$ (-f(x) + y)(g(y) + x) = (x + y + 1)(y - x - 1) $$ [i]Note: $\mathbb{R}_{>0}$ denotes the set of positive real numbers.[/i]

Suppose we have some proteins that each protein is a sequence of 7 "AMINO-ACIDS" $A,\ B,\ C,\ H,\ F,\ N$. For example $AFHNNNHAFFC$ is a protein. There are some steps that in each step an amino-acid will change to another one. For example with the step $NA\rightarrow N$ the protein $BANANA$ will cahnge to $BANNA$("in Persian means workman"). We have a set of allowed steps that each protein can change with these steps. For example with the set of steps: $\\ 1)\ AA\longrightarrow A\\ 2)\ AB\longrightarrow BA\\ 3)\ A\longrightarrow \mbox{null}$ Protein $ABBAABA$ will change like this: $\\ ABB\underline{AA}BA\\ \underline{AB}BABA\\ B\underline{AB}ABA\\ BB\underline{AA}BA\\ BB\underline{AB}A\\ BBB\underline{AA}\\ BBB\underline{A}\\ BBB$ You see after finite steps this protein will finish it steps. Set of allowed steps that for them there exist a protein that may have infinitely many steps is dangerous. Which of the following allowed sets are dangerous? a) $NO\longrightarrow OONN$ b) $\left\{\begin{array}{c}HHCC\longrightarrow HCCH\\ CC\longrightarrow CH\end{array}\right.$ c) Design a set of allowed steps that change $\underbrace{AA\dots A}_{n}\longrightarrow\underbrace{BB\dots B}_{2^{n}}$ d) Design a set of allowed steps that change $\underbrace{A\dots A}_{n}\underbrace{B\dots B}_{m}\longrightarrow\underbrace{CC\dots C}_{mn}$ You see from $c$ and $d$ that we acn calculate the functions $F(n)=2^{n}$ and $G(M,N)=mn$ with these steps. Find some other calculatable functions with these steps. (It has some extra mark.)

Find all functions $f : \mathbb{N} \rightarrow \mathbb{N}$ such that the following conditions are true for every pair of positive integers $(x, y)$: $(i)$: $x$ and $f(x)$ have the same number of positive divisors. $(ii)$: If $x \nmid y$ and $y \nmid x$, then: $$\gcd(f(x), f(y)) > f(\gcd(x, y))$$

Find the number of functions $ A\stackrel{f}{\longrightarrow } A $ for which there exist two functions $ A\stackrel{g}{\longrightarrow } B\stackrel{h}{\longrightarrow } A $ having the properties that $ g\circ h =\text{id.} $ and $ h\circ g=f, $ where $ B $ and $ A $ are two finite sets.

Find all functions $ f,g:\mathbb{Q}\longrightarrow\mathbb{Q} $ that verify the relations $$ \left\{\begin{matrix} f(g(x)+g(y))=f(g(x))+y \\ g(f(x)+f(y))=g(f(x))+y\end{matrix}\right. , $$ for all $ x,y\in\mathbb{Q} . $

Let $N$ be a positive integer whose digits add up to $23$. What is the greatest possible product the digits of $N$ can have?

Determine all functions $f:(0,\infty)\to\mathbb{R}$ satisfying $$\left(x+\frac{1}{x}\right)f(y)=f(xy)+f\left(\frac{y}{x}\right)$$ for all $x,y>0$.

Evaluate $\sum_{n=2017}^{2030}\sum_{k=1}^{n}\left\{\frac{\binom{n}{k}}{2017}\right\}$. [i]Note: $\{x\}=x-\lfloor x\rfloor$ for every real numbers $x$.[/i]

Let $\varphi$ denote the Euler phi-function. Prove that for every positive integer $n$ $$2^{n(n+1)} | 32 \cdot \varphi \left( 2^{2^n} - 1 \right).$$

Let $m_1, m_2, \cdots ,m_k$ be positive numbers with the sum $S$. Prove that \[\displaystyle\sum_{i=1}^k\left(m_i +\frac{1}{m_i}\right)^2 \ge k\left(\frac{k}{S}+\frac{S}{k}\right)^2\]

The two polynomials $(x) =x^4+ax^3+bx^2+cx+d$ and $Q(x) = x^2+px+q$ take negative values on an interval $I$ of length greater than $2$, and nonnegative values outside of $I$. Prove that there exists $x_0 \in \mathbb R$ such that $P(x_0) < Q(x_0)$.

let $ V$ be a finitive set and $ g$ and $ f$ be two injective surjective functions from $ V$to$ V$.let $ T$ and $ S$ be two sets such that they are defined as following" $ S \equal{} \{w \in V: f(f(w)) \equal{} g(g(w))\}$ $ T \equal{} \{w \in V: f(g(w)) \equal{} g(f(w))\}$ we know that $ S \cup T \equal{} V$, prove: for each $ w \in V : f(w) \in S$ if and only if $ g(w) \in S$

Given complex numbers $a,b,c$, we have that $|az^2 + bz +c| \leq 1$ holds true for any complex number $z, |z| \leq 1$. Find the maximum value of $|bc|$.